A FLAG TRANSITIVE PLANE OF ORDER 49 AND ITS TRANSLATION COMPLEMENT

The translation complement of the flag transitive plane of order 49 [Proc. Amer. Math. Soc. 32 (1972), 256-262] constructed by Rao is computed. It is shown that the flag transitive group itself is the translation complement and it is a solvable group of order 600.

The translation plane under study was constructed through a l-spread set over F (see Lemma 3.1 of if, p.258]) whose elements Li, i 49, were generated by -I
The incidence structure whose points are the vectors of F 4 Y and whose lines are Vi, 0 i 49, and their right cosets in the additive group of V with inclusion as the incidence relation is the translation plane .
From the actions of R and S, it is clear that the group G' <R,S> is transitive on the set of d.p.s of and consequently G' is flag transitive group of .
3. SPREAD SETS OF AND SOME OF THEIR PROPERTIES.
We say that a spread set over F of has a det.structure (a l,a2,a3,a4,a5,a6) if the nvmber of matrices of the spread set which are of determinant i is a i, i 6.
It may be noted that the spread set 6 of w was constructed by taking Vo,V and V 2 as the fundamental subspaces (x y,y 0 and y x respectively) and the det.strucutre of 6 is (9,9,6,8,8,8).We now construct another l-spread set 6' from 6 of w with the fundamental subspaces Vo,V38 and V 4, since the spread set is not amenable for easy computations and we study some properties of 6' and det.structures of certain matrix representative sets of .This information is useful in the computation of the translation complement G of w.I C Let T be a 4x4 matrix given by T (0 D where C-(5,6;1,3), D (4,4;5,0), I (I,0; 0,I) and 0 is the 2x2 zero matrix.Define for each LiE 6, i 49,

M.l =C+ LiD
Let ' {MilL 49}.The matt.ices Mi, 49 are ILsted in table 3. L. The entry a,b under the head[ng C.P. of M [nd[cate that the matr[ M has tle characteristic polynomial 2+a + b.It may be noted that the det.tructume of ' Ls -I (4,L2,6,12,6,8).I A,B E GL(2,F) then the det.tructume of A 'B is (4,12,6,|2,6,8) We say that the above det.structures are the allied det.structures of ' The planes associated with and ' are isomorphic and the isomorphism is given by T. Without any loss of generality we take the plane associated with ' as

PROOF.
If A is a scalar matm[x then the lemma follows tr[vlally.Conversely suppose that A-16 A {A-IMA/M E 6'} 6'.Fmom table 3.[.we notice that 6' contalus M 9 and M13 with the characteristic polynomials 2 + 3 an:[ k2 + 5 respectively and no other matrix of 6' has these polynomials as the characteristic polynomial.Therefore, we have, A-IA M 9 and A-I3A M[3.Taking A (a,b;c,d) and solving the slmu,ltaneous equations obtained from M9A AM 9 and MI3A A3 PROOF.The first part of the lemma follows fcom lemma 3.[ and the colllneation 62 when k=4 and k 21 respectively.If 6' and 6'Mk are conjugate then their det.
structures must be same and this is possible [f the det M k I. Therefore 6' -I and 6 M k are not conjugate if det M k I.
The matrices of 6' which ae of e 6' and its characteristic pol3nomlal Ls 2 + 4k + 3. The sp.ead sets 6' and 6',M 3 -'5 are not conjugate since 6' does not contain a matrix wlth the characteristic polynomial 2+ 4 +3.We reject k=41 by observing the characteristtc polynomial of M 3 and using the same argument as in the previous case.The lemma now follows.
Let M k E 6'.The det. struct,lres of 6' M k {M-Mkl.M E 6'}, are computed and are furnished in the table 3.2 for specified values of k.Thls information Ls useful in the sequel.

Let G O
be the group of all colllneatlons of that fix the d.p. 0; G0,38 be the group of all colllneations of that fix the d.p.s. 0and 38 and G0,38,4 be the group of all collineatlons that fix the d.p.s 0, 38 and 4.

A0
PROOF.Any colllneation o c G0,38,4 is of the form o--(0 A for some A c GL(2,F), satisfying the condition that for every matrix m ' there exists a matrix N E ' such that A-IMA N.That is, A-I A '.By lemma 3.1 we have A (a,0;0,a), a E F, a 0 and aalways induces a colllneation of w fixing all the d.p.s. of 7.Such a collineation ais called a scalar colllneatlon.If a is a generator of F then GO ,38,4 < > and it is of order 6.Hence the lemma.<2> and it is of order 12. LEMMA 4.2.

GO ,38
A OB) for some PROOF Any collineatlon B E G0,38 is of the form B--(0 A, B c GL(2,F).
Further A and B nmst satisfy the condition that for each matrix M ' there exists a matrix N ' such that A-IMB N. Taking M--M 4 we get a condition that A-IB '.Let A-IB M k for some k, k 49, k # 38.Then we -I obtain that the spread sets ' and 'M k are conjugate By lemma 3.2 we have k--4 and 21.
Therefore every colllneation B G0,38 either fixes the d.p. = (AB 0 D for some A,B and D E GL(2,F), satisfying the condition that for each matrix M E ' there exits a matrix N E ' such that A (B + MD) N and A B M k.That is, for every matrix M ' there exists a matrix N ' such that N-M k A-IMD.Suppose themselves.Lemma 4.1 of [I] and Theorem of[2] are now used in this paper to compute the colllneatlons of .Rao has shown that the llenar transformations on the set of d.p.s, of are R:(O,I,2 24)(25,26,27 .... ,49)
that the colllneatlon y maps the d.p.k onto the d p k' then -2 2 y c G O and maps the

Table 3 .
4or maps the d.p. 4 onto the d.p. 21and hence B either fixes the d.p. 4 or interchanges the d.p.s 4 and PROOF If yG O and maps the d.p. 38 onto the d.p.k then is of the form